Chapter III: Modeling Dynamic Transcriptional CRISPRi Circuits
3.3 Modeling Results
were taken from measurements of dCas/gRNA association rates in vitro (Mekler et al., 2016). It is unclear how closely this estimate follows in vivo kinetics. For example, the same authors show that the addition of total human lung RNA to an in vitro dCas:gRNA assembly reaction slowed dCas:gRNA binding by at least an order of magnitude. Therefore, the estimate in Table 3.1 may be an optimistic one.
Rates of association between gRNA-loaded dCas and its DNA targets have been more widely studied, but there the literature is still conflicted on their actual values.
For example, (Gong et al., 2017) report an unbinding rate of dCas from DNA of about 1/(6.5 min) in a radiolabeled pulse chase assay, but that is incompatible with the observation of (Jones et al., 2017) that dCas dissociation in vivo is driven by cell division, or with the real-time, single-molecule measurements of (Boyle et al., 2017), who could not observe sufficient unbinding events over several hours to estimate an unbinding rate for matched gRNAs (setting an upper bound on unbinding time on the order of hours). The parameters for dCas:DNA interactions chosen in Table 3.1 use binding rates from Mekler et al., and reflect the canonical understanding in the field that, for all practical purposes, dCas does not unbind from DNA.
Figure 3.2: Four examples of dynamic circuits made from CRISPRi components, simulated with the full CRISPRi model. Nodes in circuit diagrams represent gRNA expression units; blunted arrows represent dCas-mediated repression. (a) Steady states of a toggle switch for a variety of initial conditions of gRNA concentration.
(b) A 5-node oscillator. (c) A type-I IFFL (pulse generator). The purple trace tracks the IFFL output. Vertical blue and red lines mark activation and return to baseline of the input gRNA promoter, respectively. (d) Outputs of three IFFLs driven by a 5-node oscillator. When the node of the oscillator driving the IFFLs is removed (vertical red line), pulses cease. Note that not all nodes of the oscillator have corresponding visible IFFL outputs, and that the peak heights of the three IFFLs are not symmetric.
An approximation
Can we understand CRISPRi dynamics in rational, analytical terms? Should we expect an oscillator made from CRISPRi components to actually oscillate? A toggle switch? An IFFL? According to traditional genetic circuit analysis, the toggle switch (Gardner, Cantor, and J. J. Collins, 2000) and repressilators (Elowitz and Leibler, 2000) rely on cooperative binding. There is no obvious “cooperative”
mechanism in the CRISPRi model, so we might wonder whether we should expect these circuits to function at all.
Unfortunately, the full CRISPRi model outlined in Section 2.2 is not particularly amenable to analysis—even the steady state binding between a single gRNA, dCas, and the gRNA’s target is barely analytically tractable without making an unrealistic quasi-steady state assumption (finding it requires the roots of a rather messy fourth- order polynomial). To attempt to make some headway, we split the model into those parts making up an “idealized,” easy-to-analyze CRISPRi process (informally,
“first-order” considerations, though this should not be taken to imply linearity) and kinetic considerations that make CRISPRi difficult to analyze (“second-order”
considerations).
We propose the following assumptions for a first-order CRISPRi model: dCas is always present in abundance relative to both DNA targets and gRNAs; binding between gRNAs, dCas, and DNA is instantaneous; and binding of dCas to DNA targets is irreversible. Under these assumptions, the binding of gRNA to dCas to target DNA reduces to a simple “linear” model—with increasing concentrations of gRNA, dCas binds 1:1 with DNA until the DNA is completely saturated. This simplification obviously neglects some important features of CRISPRi (binding kinetics and loading effects on dCas, to name two), but it can still provide insights into how CRISPRi circuits work (or don’t).
Consider a CRISPRi toggle switch consisting of two gRNAs repressing each other.
The first-order model of the CRISPRi toggle switch can be modeled with just two differential equations:
dg1
dt = αmax(0,P1−g2)+α0min(P1,g2) −γg1, dg2
dt = αmax(0,P2−g1)+α0min(P2,g1) −γg2,
Here, g1 andg2 are concentrations of two mutually-repressing gRNAs, P1and P2 are total concentrations of promoters for those guides, α is the production rate of gRNA from an unbound promoter,α0is the production rate of gRNA from a bound promoter (leak), andγis the division rate of the cell (dilution).
Under what conditions does this system admit two stable steady states? To answer this, we should consider the intermediate steady state of the system, far from the
bounds set by 0, P1, and P2. In general, toggle-switch-like circuits undergo a supercritical pitchfork bifurcation at this point. When it is stable, the system admits only one state (Figure 3.3A), but when it is unstable, the system will have two steady states (the “togglable” steady states) (Figure 3.3B). In particular, the middle steady state will be unstable (and the toggle switch will correctly “toggle”) if and only if that system has a single non-trivial steady state that is unstable. This corresponds to the case where at least one of the eigenvalues of the Jacobian of the system has positive real part. To find when this is true, we note that far from any saturating bounds (where we are likely to find the central steady state), the system reduces to
Figure 3.3: Flow fields for the first-order approximation model of a CRISPRi toggle switch. Depending on the parameters chosen, the toggle could (a) admit two stable steady states separated by an unstable steady state or (b) admit a single stable steady state.
dg1
dt = αP1g2(α−α0) −γg1, dg2
dt = αP2g1(α−α0) −γg2,
whose Jacobian has eigenvalues−(α−α0) −γand(α−α0) −γ. The first eigenvalue always has negative real part. The first eigenvalue has positive real part (and the system “toggles”) whenα−α0 > γ. In short, a toggle should function as long as the difference between production rates of bound and unbound promoters is sufficiently large relative to dilution.
We can apply a similar analysis to a three-node CRISPRi repressilator, which is an oscillator consisting of an odd number of guide RNAs in a circular circuit topology, each gRNA repressing the next in the cycle (similar to Figure 3.2B). Bounded dynamical systems with repressilator-like architecture typically have a single non- trivial steady state. As with the toggle, the desired behavior (oscillations, in this case) can occur only when that central steady state is unstable. In principle, an unstable central steady state is not sufficient to guarantee oscillations; in practice, molecular species concentrations are bounded by dilution, and there are no other possible stable steady states to the repressilator system, which leaves little room for non-oscillitory (or chaotic) behavior.
The Jacobian for a three-node CRISPRi repressilator has eigenvalues
1 2
±p
−3(α−α0)2+(α−α0) −2γ
and −(α− α0) − γ. The last eigenvalue al- ways has negative real part. The first pair of eigenvalues each have positive real part (and the system oscillates) whenα−α2 0 > γ. As with the toggle switch, the difference between production rates of bound and unbound promoters must be sufficiently great for the CRISPRi repressilator to oscillate.
We will soon see that this simplified model is a poor predictor of exactly what behavior a CRISPRi circuit with particular parameters or will not exhibit. This is beside the point; what we learn from the simplified CRISPRi model is that cooperativity is notnecessary for either a toggle switch or a repressilator, despite classical understanding in the literature (Gardner, Cantor, and J. J. Collins, 2000;
El-Samad, Vecchio, and Khammash, 2005). Cooperativity, it seems, is necessary only when genes are expected to bind in a Hill-like fashion. Perfectly linear binding with sharp saturation, as we should expect in CRISPRi, is another perfectly viable path to useful instability.
Note that the result that a CRISPRi oscillator does in fact oscillate contradicts the modeling results by (Santos-Moreno, Tasiudi, et al., 2020). In that work, the authors model a CRISPRi toggle as a slightly more tractable proxy for the oscillator. The authors analyze this model using BioSWITCH, a toolbox for limit point detection and bistability analysis across the space of all “reasonably” bounded parameters.
Figure 3.4: Steady-state analysis with the BioSWITCH toolbox shows that bistability is possible with dCas:gRNA complex dilution (right), but not without it (left). Plots show possible steady-state values of gRNA #1 as a function of one parameter, with the others held fixed at values computed to optimally detect bifurcation points.
The authors show that a simple model similar to our full model fails to predict the steady-state instability required for bistability in the toggle (Figure 3.4, left). The authors additionally show that bistability can be recovered by adding off-target and cross-target binding.
This analysis contradicts ours in two ways. Firstly, our model predicts that a CRISPR toggle switch should function even without any off-target or cross binding.
Secondly, our model predicts that off-target binding shoulddecreasethe robustness of a CRISPRlator, not increase it (Figure 3.5A).
One difference between our model and that of Santos-Moreno and Taisiudi is that our model includes degradation of gRNA-bound dCas complexes. Adding these degradation reactions to their model is sufficient to allow steady-state instability ac- cording to BioSWITCH (Figure 3.4A). Our own model shows that with gRNA:dCas dilution removed, the toggle switch’s “on” steady state becomes unstable, causing the high-state gRNA to increase without bound (Figure 3.4B).
Now that we have some theoretical justification for believing that a CRISPRi toggle
Figure 3.5: Addition of off-target binding destroys oscillations in the 3-node CRISPRlator, contra Santos-Moreno, Taisiudi, et al., 2020. In the top row, ad- ditional non-target binding sites have been added. In the bottom row, additional non-target binding sites have been added and guides can bind at a low rate to mismatched target sites.
switch or repressilator should be possible to build, we will explore what conditions allow those circuits to function under the full CRISPRi model.
Repressilators can be made with CRISPRi, but they can display strong initial condition dependence
Consider, again, the 3-node CRISPRi repressilator, or CRISPRlator. Our model predicts it to be quite slow, with a period of about seven hours (about three times as long as the original protein-based repressilator (Elowitz and Leibler, 2000)). This is because where the time scale of the repressilator is limited by active protein degradation, the timescale of the CRISPRlator is set by dilution.
Interestingly, even under parameterizations that allow oscillations, the CRISPRlator does not oscillate for all initial conditions (Figure 3.6). It is possible for the 3-node CRISPRlator to possess both a stable limit cycle and an unstable limit cycle inside the stable limit cycle which screens off trajectories with insufficient differences in concentrations of different gRNAs. These latter trajectories, which fall inside the unstable limit cycle, spiral to a stable steady state.
Note that this behavior differs from the behavior of the classic repressilator, which (in its reduced three-species form) can only have up to a single, stable limit cycle (V. P. Golubyatnikov and Ivanov, 2018; Likhoshvai, Vladimir P. Golubyatnikov, and Klebodarova, 2020). Also note that the existence of multiple limit cycles in the CRISPRlator implies that linearization characterization of the steady state of
a CRISPRlator is insufficient to determine whether the CRISPRlator will actually oscillate. If the CRISPRlator’s steady state is unstable, then it will oscillate; but if the steady state isstable, then it could either have a pair of stable/unstable limit cycles (in which case it can oscillate), or no limit cycles (in which case it will not). Accordingly, when we computationally screen for oscillations in various CRISPRlators, as in Figure 3.7, we do so using numeric simulation and heuristic oscillation detection rather than with stability analysis.
A)
B)
Two identical CRISPRlators with slight differences in initial condition
Initial [gRNA #2] = 15 nM
Multiple limit cycles in a 3-node CRISPRlator
Initial [gRNA #2] = 17 nM
Figure 3.6: (a)The same CRISPRlator (with the same rate parameters) oscillates for some initial conditions but not for others. In both cases, dCas begins at 43 nM and all other non-DNA species other than gRNA #2 start at 0 nM.(b)Trajectories for a variety of initial conditions reveal multiple nested limit cycles in the 3-node CRISPRlator under some parameterizations. In this example, an unstable limit cycle (dotted red loop) screens trajectories with too little total gRNA from a stable limit cycle (solid blue loop).
The CRISPRlator is fragile
Unfortunately, the CRISPRi repressilator is fragile, and appears to sit close to a bifurcation in parameter space. It is, for example, fairly sensitive to dCas production rate, and ceases to oscillate with less than about 10% or more than about 150% of
the default dCas production rate in Table 3.1. This indicates that dCas production levels may have to be fine-tuned specifically for any particular CRISPRi circuit.
Luckily, this is a relatively easy parameter to tune in a real cell.
The repressilator is also not generally robust against transcriptional leak. The first- order model predicts that an increase in leak should stabilize the system towards a steady state, eventually driving it to equilibrium with no oscillations. This is reflected in the full model, and not only for the CRISPRlator—addition of as little as 1% leak destroys oscillations in all but one of the simulations shown in Figure 3.2 (the exception is the CRISPRlator-driven triple IFFL of Figure 3.2D, which breaks between 2 and 4% leak).
That a CRISPRlator has, in fact, been built successfully suggests that CRISPRi might, in fact, have extremely low leak in the true sense of allowing transcription while the repressor is bound (Santos-Moreno, Taisiudi, et al., 2020; Kuo et al., 2020). Repression with dCas has been reported in cell-free extract with fold- repression between 7 and 100, and in vivo with similar repression strengths, which puts the best CRISPRi repression in a leak range that should not be likely to allow a repressilator to work. Our model of the CRISPRlator suggests that the “leak”
observed in these experiments may be a function ofslow binding, not failure of a bound repressor—that is, observed “leak” occurs because a promoter is not bound some of the time, not because a promoter is bound to dCas and produces transcripts anyway.
Active degradation can offset leak-based circuit fragility
There are a few different knobs we can turn to make the CRISPRi repressilator more robust to transcriptional leakiness. We can decrease the rate of production of either dCas or gRNAs; we can speed the binding between dCas:gRNA complexes and DNA (in contrast, speeding binding between dCas and gRNAs appears to have little effect); we can add active degradation of dCas (also increasing the speed of the oscillator considerably); and we can grow the repressilator from three nodes to five nodes.
As an example, let us consider the interaction between dCas degradation rate and leak rate. Figure 3.7 shows the performance of simulations of several circuits as a function of the degradation rate and leak rate parameters, with other parameters as shown in Table 3.1. We can think of these charts as a sort of two-dimensional
“specification sheet” for dCas to allow different circuits to function properly. For any particular leak rate, the circuit will either oscillate (Figure 3.7A, B, D, E, and F) or toggle (Figure 3.7C) only when dCas is degraded at a rate within a proper range. Too much dCas degradation will destroy all examined circuits, and a minimum amount of degradation is required for some. Notably, the “correct”
dCas degradation specification is different for different circuits (compare Figures 3.7A, 3.7B, and 3.7C), dCas expression rates (Figures 3.7D and E), and background activity (different numbers of oscillators in Figure 3.7F). Interestingly, the toggle switch has similar parameter requirements on these two axes, suggesting that there
may be some requirements shared by some interesting class of dynamic CRISPRi circuits.
We can produce a similar “specification sheet” for dCas degradation and leak for a 5-node CRISPRi repressilator, as shown in Figure 3.7B. The 5-node repressilator is more robust than the 3-node oscillator. Indeed, the 5-node repressilator can operate with as much as 10% leak or as little as no dCas degradation at all. In the case of CRISPRi repressilators, bigger is not only better, but potentially easier.
On the other hand, the parameter requirements of the toggle switch appear to be quite similar to those of the 3-node repressilator (Figure 3.7C). Admittedly, the toggle switch and 3-node repressilator have very similar architecture, but the fact that both circuits require similar degradation rates and minimum promoter leak suggests that the regime of functional repressilators may have not-yet-understood underlying properties that are broadly useful for constructing CRISPRi circuits.
There are more than two tunable knobs in the CRISPRi system. One that we have already seen to be important is theproductionrate of dCas. Figure 3.7D shows how the target parameter set changes with different levels of dCas production. The good news is that with low enough dCas expression there is no need for dCas degradation (though with dCas steady state levels that low, stochastic fluctuations become a more serious problem). The bad news here is that at least oneengineerablebut notreadily tunableparameter of CRISPRi (namely, dCas degradation rate) has acceptable value ranges that don’t overlap for some choices of dCas production rate. This should not be too much of a problem for making a single repressilator, but it does complicate the design and integration of multiple CRISPRi circuits in the same cell. For example, Figure 3.7E shows the expected effect of expressing two identical CRISPRi repressilators in parallel with no directly cross-interacting nodes. The increased load on dCas drops the effective steady-state concentration of dCas as perceived by each individual oscillator, which has a similar effect as dropping dCas production rate.
Namely, this shifts the required rate of dCas degradation. A repressilator that works on its own can be expected to fail when a second repressilator is added, unless dCas’s degradation rate is exquisitely well-tuned. More generally, it seems likely that different circuits may require dCas variants with different degradation rates.
Reciprocally, we could tune degradation rate to compensate for changes in other parameters. The CRISPRlator requires dCas to be produced at a tuned rate—too much or too little dCas production will destroy the circuit’s function. Changing the rate at which dCas is degradedalsochanges those dCas production rate requirements, potentially allowing degradation to compensate for any lack of control over dCas concentration (Figure 3.7D).
Finally, we can consider the robustness of the CRISPRlator to differences in the strengths of the promoters driving gRNA production. In general, repressilators require nodes with roughly similar repression strengths. The more sensitive the CRISPRlator to gRNA promoter strengths, the more difficult a CRISPRlator will be to engineer. Fortunately, as shown in Figure 3.8, the 5-node CRISPRlator is robust to (most) changes of at least 10-fold in two adjacent gRNA.
Figure 3.7: Acceptable degradation rates vary by circuit architecture and parame- terizations. Leak is given in units of the rate of gRNA production from an unbound promoter. Degradation rate is in units of cell division rate. Colors at each point in parameter space indicate whether the circuit oscillates (blue) or not (red) for those given parameters. Results are given fora)a 3-node CRISPRlator,ba 5-node oscil- lator, andc)a CRISPRi toggle switch. Inc, Colors indicate the separation distance between the two gRNAs at steady state; red indicates no detectable bistability. d) Changes in degradation requirements for a 3-node CRISPRlator with different lev- els of dCas expression. Areas under each curve represent parameters for which the circuit oscillates. Production rates are given in units of min−1 (the “default” dCas speed used in (textbfa)). e)Changes in dCas expression requirements for the 3-node CRISPRlator with different rates of dCas9 degradation. The shaded region repre- sents production rates that admit oscillations. Oscillations could not be recovered with any higher degradation rate. Transcriptional leak is set to 0. f)Parameters for which different numbers of independent 3-node CRISPRlators oscillate while oper- ating in the same cell. Larger number of oscillators become steadily less tolerant of both degradation and leak.